Mathematician Tim Pennings watched his dog Elvis fetch balls thrown in the water and noticed that the dog consistently chose the quickest route. Running is faster than swimming, so the overall time the dog spends heading to the ball depends on how the dog decides to split his path into running and swimming parts.

Elvis could run directly into the water and swim a long way to the ball, which would mean traveling the shortest distance, but not getting there as fast as possible. Another possibility is to run on the sand until he is even with the ball, and then swim to it. A third option is to run part of the way along the shore and then finish traveling to the ball by swimming in the water. Elvis always chose this last option, which resulted in reaching the ball the fastest. Mathematicians describe his actions by saying that Elvis optimized his travel time.

With information about the position of the ball and the dog, and the dog’s running and swimming speeds, it is possible to use calculus to determine the exact place at which the dog should switch from running to swimming in order to minimize his travel time. Pennings has suggested that dogs do in fact know calculus[3], because their paths match what the mathematics of calculus predict.

I think it’s more accurate to say that dogs act as though they know calculus rather than to say that they actually know calculus. It’s a small, but important distinction. I agree that dogs act to optimize their travel time when fetching in the water—I’ve observed dogs doing this—but that does not mean they are making complex mathematical calculations. It’s more likely that their experience allows them to make choices that result in getting to the ball faster.

Watching dogs fetch in an optimal way is no less remarkable to me than if they were using calculus. Have you seen dogs performing the kind of behavior that led Pennings to suggest they know calculus?

Karen B. London, PhD, is a Certified Applied Animal Behaviorist and Certified Professional Dog Trainer whose clinical work over the last 17 years has focused on the evaluation and treatment of serious behavioral problems in dogs, especially aggression. Karen has been writing the behavior column for The Bark since 2012 and wrote The Bark’s training column and various other articles for eight years before that. She is an adjunct professor in the Department of Biological Sciences at Northern Arizona University, and teaches a tropical field biology course in Costa Rica. Karen writes an animal column, The London Zoo, which appear in The Arizona Daily Sun and is the author of five books on canine training[4] and behavior. She is working on her next book, which she expects to be published in 2017.